Usually, the upper shock bushings get squeaky. Couldn't hurt to grease the control arm bushings either. '71 should still have grease fittings on them.
Nope never seen one from the factory that had fittings installed and as I said there isn't enough space to install a std fitting... I've owned two Mavericks and probably a dozen '67-'71 Falcons, Fairlanes & Torinos none had factory fittings... At one time there was a special "skinny" fitting that would slip into the opening but I haven't seen those in years... Most just had the spring/shock tower cut or torched for access...
Yup,I remember many '60s Ford Mustangs had this problem. I believe the Maverick had the same suspension. As I remember, the upper control arm pivoted on a shaft. There were two pivots, one towards the front and one towards the rear, tucked up in the spring/shock tower. There were usually grease fitting plugs on each end of the shaft (one facing forward, one facing back). It was a tight fit to the spring/shock tower sheetmetal. I remember seeing a lot of rudely torched holes in the spring/shock tower to gain access to the grease fitting plugs. The hole would typically be a little over an inch in diameter. The plugs would then be replaced with grease fittings to allow lubrication of the control arm pivots and stop the squeaking (at least temporarily, they had to be greased regularly).
Come to think of it, I remember a special tool that could be used to grease the control arms. But you had to remove the plug, install the tool, grease the pivot, remove the tool and reinstall the plug. It was a bit of PIA to use. Maybe that was the special "skinny" fitting you mention.